Related papers: Bounding the Interleaving Distance for Mapper Grap…
Mapper graphs are widely used tools in topological data analysis and visualization. They can be understood as discrete approximations of Reeb graphs, providing insight into the shape and connectivity of complex data. Given a…
The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and…
The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has widely been used in applications, however its use on real data means that it is desirable and increasingly necessary to…
Merge trees, contour trees, and Reeb graphs are graph-based topological descriptors that capture topological changes of (sub)level sets of scalar fields. Comparing scalar fields using their topological descriptors has many applications in…
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for…
Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric…
The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show…
One of the prevailing ideas in geometric and topological data analysis is to provide descriptors that encode useful information about hidden objects from observed data. The Reeb graph is one such descriptor for a given scalar function. The…
In this work we study the interleaving distance between merge trees from a combinatorial point of view. We use a particular type of matching between trees to obtain a novel formulation of the distance. With such formulation, we tackle the…
Merge trees are a type of graph-based topological summary that tracks the evolution of connected components in the sublevel sets of scalar functions. They enjoy widespread applications in data analysis and scientific visualization. In this…
Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such…
Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the…
Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric…
A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two…
Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite…
In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we…
Interleaving distances provide a fundamental tool for comparing persistence modules and have been widely used in topological data analysis. Their definitions are typically based on translation structures (shift operations) on the indexing…
In this paper, we present a new metric distance for comparing two large graphs to find similarities and differences between them based on one of the most important graph structural properties, which is Node Adjacency Information, for all…
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit…
Graphs are versatile tools for representing structured data. As a result, a variety of machine learning methods have been studied for graph data analysis. Although many such learning methods depend on the measurement of differences between…